A certain text contains 1/3 vowels and 2/3 consonants. (a) A letter is selected

Question

A certain text contains 1/3 vowels and 2/3 consonants.

(a) A letter is selected at random from this text, and you’re to guess whether the letter is vowel or consonant. Let p be the probability that you guess the letter is consonant. What is the optimal value of p that leads you to a correct guess?

(b) If five letters are selected at random, and for each letter you guess vowel with probability 1/3 and consonant with probability 2/3, what is the probability that exactly two of your guesses are correct (and 3 guesses are incorrect)?

Explanation / Answer

p(vowel)=1/3

p(consonant)=2/3

a) 2/3 should be the optimal value of p

b) this is given as 5C2*(1/3)^2*(2/3)^3 =((5*4)/2)*(1/3)^2*(2/3)^3 =0.329

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